Maximal escape rate for shifts

Claudio Bonanno; Giampaolo Cristadoro; Marco Lenci

doi:10.3934/dcds.2022135

Article Contents

2022, Volume 42, Issue 12: 6007-6029. Doi: 10.3934/dcds.2022135

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Maximal escape rate for shifts

1.
Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
2.
Dipartimento di Matematica e Applicazioni, Università di Milano - Bicocca, Via Roberto Cozzi 55, 20125 Milano, Italy
3.
Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy
4.
Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy

^*Corresponding author: Claudio Bonanno
^*Corresponding author: Claudio Bonanno

Received: December 2021

Revised: August 2022

Early access: September 2022

Published: December 2022

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Abstract

We consider the shift transformation on the space of infinite sequences over a finite alphabet endowed with the invariant product measure, and examine the presence of a hole on the space. The holes we study are specified by the sequences that do not contain a given finite word as initial sub-string. The measure of the set of sequences that do not fall into the hole in the first $ n $ iterates of the shift is known to decay exponentially with $ n $, and its exponential rate is called escape rate. In this paper we provide a complete characterization of the holes with maximal escape rate. In particular we show that, contrary to the case of equiprobable symbols, ordering the holes by their escape rate corresponds to neither the order by their measure nor by the length of the shortest periodic orbit they contain. Finally, we adapt our technique to the case of shifts endowed with Markov measures, where preliminary results show that a more intricate situation is to be expected.

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Mathematics Subject Classification: Primary: 37B10, 37A05, 37E05.

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Figure 1. The escape rates $ \gamma_{{w}} $ for $ p\in [\frac 12,1] $, for all holes of length $ r = 4 $

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Figure 2. The relative error between a very precise numerical approximation of $ \gamma^r_{max} $ and the lower bound $ l_b $ in Corollary 3.9, defined by $ RE(r): = (\gamma^r_{max} - l_b)/ \gamma^r_{max} $ and displayed as a function of the length $ r $ of the hole in log-linear scale. The decay towards zero shows that the accuracy of the estimate improves exponentially with the length of the hole. Different curves correspond to different values of $ p $ (from bottom to top: $ p = 0.85 $, $ p = 0.9 $, $ p = 0.95 $)

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Figure 3. The escape rates for the system of Section 4, as functions of $ \pi_{aa},\pi_{bb}\in (0,1) $: the blue graph is for the holes $ w = (aaa) $ and $ w = (bbb) $; the red graph is for the holes $ w = (aab) $, $ w = (bba) $, $ w = (baa) $ and $ w = (abb) $; the green graph is for the holes $ w = (aba) $ and $ w = (bab) $

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